Prove 1 + 2 + 3 + β¦β¦. + n = (π§(π§+π))/π for n, n is a natural number
Step 1: Let P(n) : (the given statement)
Let P(n): 1 + 2 + 3 + β¦β¦. + n = (n(n + 1))/2
Step 2: Prove for n = 1
For n = 1,
L.H.S = 1
R.H.S = (π(π + 1))/2 = (1(1 + 1))/2 = (1 Γ 2)/2 = 1
Since, L.H.S. = R.H.S
β΄ P(n) is true for n = 1
Step 3: Assume P(k) to be true and then prove P(k + 1) is true
Assume that P(k) is true,
P(k): 1 + 2 + 3 + β¦β¦. + k = (π(π + 1))/2
We will prove that P(k + 1) is true.
P(k + 1): 1 + 2 + 3 +β¦β¦. + (k + 1) = ((k + 1)( (k + 1) + 1))/2
P(k + 1): 1 + 2 + 3 +β¦β¦.+ k + (k + 1) = ((π€ + π)(π€ + π))/π
We have to prove P(k + 1) is true
Solving LHS
1 + 2 + 3 +β¦β¦.+ k + (k + 1)
From (1): 1 + 2 + 3 + β¦β¦. + k = (π(π + 1))/2
= (π(π + π))/π + (k + 1)
= (π(π + 1) + 2(π + 1))/2
= ((π + π)(π + π))/π
= RHS
β΄ P(k + 1) is true when P(k) is true
Step 4: Write the following line
Thus, By the principle of mathematical induction, P(n) is true for n, where n is a natural number

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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